Theoretical analysis of the leptonic decays $B\to \ell \ell {\ell }^{\prime }{\bar{\nu }}_{{\ell }^{\prime }}$

Abstract

We discuss the amplitude of the $B\to l^{+}{l}^{-}{l}^{\prime }{\nu }^{\prime }$ decays and the differential decay rate $d^2\Gamma /d{q}^{2}d{q}^{\prime 2}$, $q$ the momentum of the $l^{+}{l}^{-}$ pair emitted from the electromagnetic vertex and $q^{\prime }$ the momentum of the $l^{\prime }{\nu }^{\prime }$ pair emitted from the weak vertex. For the relevant form factors, we construct dispersion representations in $q^2$ which consistently take into account the Ward identity constraints at $q^2=0$ and the contributions of light vector resonances. This allows a reliable description of the form factors in the range $0<q^2\le 1$ GeV${}^2$ that saturates around 99% of the decay rate. The differential decay rate behaves at small $q^2$ as $d\Gamma (B\to l^{+}{l}^{-}{l}^{\prime }{\nu }^{\prime })/d{q}^2\propto 1/q^2$ in the limit $m_l^{\prime }=0$, but contains also more singlular contribution of order $m_l^{\prime 2}/q^4$, which we take into account. For the case $m_l^{\prime }\le m_l$, the latter may be neglected and one obtains a mild logarithmic dependence of $\Gamma (B\to l^{+}{l}^{-}{l}^{\prime }{\nu }^{\prime })$ on $m_l$. For the case $m_l\ll m_l^{\prime }$, however, the $m_l^{\prime 2}/q^4$ terms dominate the decay rate leading to $\Gamma (B\to l^{+}{l}^{-}{l}^{\prime }{\nu }^{\prime })\sim m_l^{\prime 2}/m_l^2$. We find the following features of the four-lepton $B$-decays: (i) The decay rates $\Gamma \left(B\to {\mu }^{+}{\mu }^{-}(\mu {\nu }_{\mu },e{\nu }_e)\right)$ are fully dominated by the region of light vector resonances $q^2\simeq M_{\rho }^2,M_{\omega }^2$; (ii) The decay rate $\Gamma (B\to e^{+}{e}^{-}e{\nu }_e)$ receives comparable contributions from the region near $q^2\sim 4m_e^2$ and from the resonance region; (iii) One finds a strong enhancement of the decay rate $\Gamma (B\to e^{+}{e}^{-}\mu {\nu }_{\mu })\sim m_{\mu }^2/m_e^2$ which is dominated by the region $q^2\sim 4m_e^2$ due to the terms $O(m_{\mu }^2/q^4)$ in the differential distribution.

Comments: 14 pages. The branching fractions for the leptonic B decays into the final states containing identical leptons have been updated. Version to appear in Physical Review D